In normal circumstances, they can appear as part of a warp effect, or spawn from the altars of an Eldritch Shrine (aka Obelisk). Inserting eldritch eyes into the altar capstones will increase their spawn rate, and some shrines naturally spawn Eldritch Guardians regardless of placed Eldritch Eyes. Either way, they bring with them a mysterious and far-reaching fog to hide within (all nearby players get a blindness-like effect). However, eldritch guardians which appear in familiar worlds (not just the Overworld, but even the Nether and End) are visibly transparent, and at a distance fade to invisibility in their fog. This reflects their weakness in our dimensions, which are as hostile to them as theirs is to us. Only in their home, the fabled Outer Lands, do they achieve their full strength, gaining better stats and a Runic Shield.
Eldritch Guardians have a vision range of 40 or so, but their fog extends outward to more than half that (23). They have 50 health (25 Hearts). If within 9 blocks of the player, they will always try to close for melee, doing 4 damage (2 hearts). This is a fairly useful habit, as they are more than 2 blocks tall, and can be kept away by the same 2-block ceiling used for Endermen. From a distance, they will normally fire a dark sphere of unholy magic, which also does 4 damage, but can also inflict 7 seconds of Weakness I. Sometimes the dark sphere will be replaced by a "screech" -- the guardian fires a dark beam which is pretty much undodgeable, though it cannot pass through blocks. This beam does only 1 damage immediately, but also inflicts Wither I for 20 seconds. The screech and perhaps the dark ball can also inflict temporary or regular warp on the victim.
They normally drop ethereal essence (Alienis, Examinis, and (formerly a bugged "untyped" version)). They also have a rare drop, the mysterious "eldritch eye", which can be used to activate an Eldritch Shrine. They will automatically target and attack Crimson Knights and Crimson Clerics (and vice versa) but will nearly always lose even when only against one knight or cleric.